I've written a short blog discussing the idea of extending Binary Strings with DP. It also discusses the solution to problem E: Counting Binary Strings from last Div 2 round.

To help you solidify the concepts discussed, I've created 5 + 1 practice problems. You can access them here: https://mirror.codeforces.com/group/7Dn3ObOpau/contest/498710

https://cfstep.com/training/tutorials/strings/extending-binary-strings/

The problems are untested. If you see any issues, please let me know.